Let pR rightarrow R be a polynomial function Let subset R be

Let p:R rightarrow R be a polynomial function. Let subset R be bounded. Let B = p(A). Prove that B is bounded. Follow these steps. If B is unbounded, there is a sequence (x_n) in A such that |p(_n)| > n, for all n element of N. There exists a subsequence (x_n_k) that converges to a number x*. Show that p(x_n_k) rightarrow p(x*). Explain why this leads to a contradiction and conclude that B is bounded.

Solution

Given B=P(A)

Also, if B is unbounded there is a sequence in A such that |p(x_n)| > n

For example,

Let sequence x_n = x₁ + x₂ +x₃ +...+x_n

Another sequence x_nk = x_1k + x_2k +...+x_nk =x^*

this sequence converges to a value x^*. and so x^* is an element of A.

if x^* element of A, then x_nk element of A. because x^*=x_nk

so p(x_nk)= p(x^*)

since p(A)=B, p(x^*)=p(x_nk) is element of B.

But given that |p(x_n)| >n ,for all n element of N.

so |p(x_n)| is approximatly equal to infinity. since its value is greater than every natural number.

p(x_n) cannot converge to aparticular value and so p(x_n) is not an element of B.

(Function p(A) has a value for every value of A.but p(x_n) does not converge to a value but it diverges to infinity.so p(x_n) is not in B. and x_n is not a sequence in A) and hence B is bounded.